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Finite-difference approximation of the boundary conditions of the second and third order for the nonlinear heat conduction equation

Posted on: 22.08.2012

Let us consider non-linear heat conduction equation

, (1)

where, (2)

with the initial condition , and boundary conditions , (3)

At . If , condition (3) is the boundary condition of the second order, but if , it is the boundary condition of the third order.

Let us introduce analytical grids in space and time:, , , .

Then, define grid functions , , , , , , , .

With we have, . In order to find при , let us use implicit difference scheme (about explicit scheme see remark 2 listed below),
where .It is a generalization of the introduced scheme 1 for the coefficients equation not dependent on x. Suppose that partial derivatives , re bounded and functions

(4)

are distributed at the whole field by Taylor formula with remainder terms , and respectively. Using terms 2 and carrying out arguments similar to the arguments listed below, we can prove approximation scheme error .

To approximate the heat flow in the left boundary condition (3) right difference derivative can be used. (5)

Meanwhile approximation error would have had just first sequence in space.

Universal way to improve this procedure is the introduction of phantom (ghost, imaginary, fictitious) node [2, 3] out of modeling area (see Fig. 1) and the use of central difference derivative .

Fig. 1.Phantom nodex_1

It is also possible to introduce approach based on the account of heat equation for the boundary node. In [4], this approach is applied to the linear heat equation with constant coefficients. Below it is generalized in case of nonlinear heat equation.

Fix the moment of time , . Rewrite condition (3), using the last notations (4):

Let us derivate difference approximation and for their further substitution in (7).

By definition. (8)

By Taylor formula , whence .It is equal to the fact, that approximates factor into (8) with error . At the same time for a second factor it is done . Using limitation previously assumed and limitation K (T),specified in clause (2), from the equality we have

(9)

Only approximation is left. By definition . Using heat conduction equation (1), replace the right side of this equation:

. (10)

By analogy with derivation from (8) equality (9), we obtain from (10) the equality

. (11)

Substitute approximations (9) and (11) into equality (7), and the result of this substitution – into boundary condition (6). Neglecting terms of order , finally we have

. (12)

At this coincides with [4].

Rewrite condition (12) as

(13)

Remark 1. From (13) it is obvious, that coefficient at exceeds the absolute value at .It ensures strict diagonal dominance [5] of difference equations matrix system.

Remark 2. Approximation (13) gives the best fit to using with an explicit difference scheme. It is also possible by analogy with [4] use the ratio. (14)

Dear Daniel, these non-standard-looking formulae are obtained via the approach, described in "The Theory of Difference Schemes" by Alexander A. Samarski, pp. 82 -- 84.
The equation is linearized in the usual way (for explicit finite-difference schemes): one takes the values of temperature-dependent coefficients from the previous time level. The stability issue is managed using sufficiently small time step (see method of frozen coefficients).
As for numerical experiments, we have carried out a lot. The results show that for thin space grids the described boundary condition approximation (of second order in space) is much more effective than the simple one (of first order).